ABSTRACT We provide a framework for online con∞ict-free coloring (CF-coloring) of any hyper- graph. We use this framework to obtain an e-cient randomized online algorithm for CF-coloring any k-degenerate hypergraph. Our algorithm uses O(klogn) colors with high probability and this bound is asymptotically optimal for any constant k. Moreover, our al- gorithm uses O(klogklogn) random bits with high probability. As a corollary, we obtain asymptotically optimal randomized algorithms for online CF-coloring some hypergraphs that arise in geometry. Our algorithm uses exponentially fewer random bits compared to previous results. We introduce deterministic online CF-coloring algorithms for points on the line with respect to intervals and for points on the plane with respect to halfplanes (or unit discs) that use £(logn) colors and recolor O(n) points in total.

[Show abstract][Hide abstract]ABSTRACT: A con∞ict-free coloring for a given set of disks is a coloring of the disks such that for any point p on the plane there is a disk among the disks covering p having a color difierent from that of the rest of the disks that covers p. In the dynamic o†ine setting, a sequence of disks is given, we have to color the disks one-by-one according to the order of the sequence and maintain the con∞ict-free property at any time for the disks that are colored. This paper focuses on unit disks, i.e., disks with radius one. We give an algorithm that colors a sequence of n unit disks in the dynamic o†ine setting using O(logn) colors. The algorithm is asymptotically optimal because ›(logn) colors is necessary to color some set of n unit disks for any value of n (8).

[Show abstract][Hide abstract]ABSTRACT: In a coloring of a set of points P with respect to a family of geometric regions one requires that in every region containing at least two points from P, not all the points are of the same color. Perhaps the most notorious open case is coloring of n points in the plane with respect to axis-parallel rectangles, for which it is known that O(n0.368)O(n0.368) colors always suffice, and Ω(logn/log2logn) colors are sometimes necessary.In this note we give a simple proof showing that every set P of n points in the plane can be colored with O(logn) colors such that every axis-parallel rectangle that contains at least three points from P is non-monochromatic.

[Show abstract][Hide abstract]ABSTRACT: Some of the routing protocols used in telecommunication networks route traffic on a shortest path tree according to configurable integral link weights. One crucial issue for network operators is finding a weight function that ensures a stable routing: when some link fails, traffic whose path does not use that link should not be rerouted. In this paper we improve on several previously best results for finding small stable weights. As a conceptual contribution, we draw a connection between the stable weights problem and the seemingly unrelated unique-max coloring problem. In unique-max coloring, one is given a set of points and a family of subsets of those points called regions. The task is to assign to each region a color represented as an integer such that, for every point, one region containing it has a color strictly larger than the color of any other region containing this point. In our setting, points and regions become edges and paths of the shortest path tree, respectively, and based on this connection, we provide stable weight functions with a maximum weight of O(nlogn) in the case of single link failure, where n is the number of vertices in the network. Furthermore, if the root of the shortest path tree is known, we present an algorithm for determining stable weights bounded by 4n, which is optimal up to constant factors. For the case of an arbitrary number of failures, we show how stable weights bounded by 3 n n can be obtained. All the results improve on the previously best known bounds.

Data provided are for informational purposes only. Although carefully collected, accuracy cannot be guaranteed. The impact factor represents a rough estimation of the journal's impact factor and does not reflect the actual current impact factor. Publisher conditions are provided by RoMEO. Differing provisions from the publisher's actual policy or licence agreement may be applicable.

Page 1

Online conflict-free colorings for hypergraphsAmotz Bar-Noy∗Panagiotis Cheilaris†Svetalana Olonetsky‡Shakhar Smorodinsky§AbstractWe provide a framework for online conflict-free coloring (CF-coloring) of any hyper-graph. We use this framework to obtain an efficient randomized online algorithm forCF-coloring any k-degenerate hypergraph. Our algorithm uses O(klogn) colors with highprobability and this bound is asymptotically optimal for any constant k. Moreover, our al-gorithm uses O(klogklogn) random bits with high probability. As a corollary, we obtainasymptotically optimal randomized algorithms for online CF-coloring some hypergraphsthat arise in geometry. Our algorithm uses exponentially fewer random bits compared toprevious results.We introduce deterministic online CF-coloring algorithms for points on the line withrespect to intervals and for points on the plane with respect to halfplanes (or unit discs)that use Θ(logn) colors and recolor O(n) points in total.1 IntroductionA hypergraph is a pair (V,E), where V is a finite set and E ⊂ 2V. The set V is called theground set or the vertex set and the elements of E are called hyperedges. A proper k-coloring ofa hypergraph H = (V,E), for some positive integer k, is a function χ : V → {1,2,...,k} suchthat no S ∈ E with |S| ≥ 2 is monochromatic. Let χ(H) denote the minimum integer k forwhich H has a k-coloring. χ(H) is called the chromatic number of H. A conflict-free coloring(CF-coloring) of H is a coloring of V with the further restriction that for any hyperedgeS ∈ E there exists a vertex v ∈ S with a unique color (i.e., no other vertex of S has thesame color as v). Both proper coloring and CF-coloring of hypergraphs are generalizations ofvertex coloring of graphs (the definition coincides when the underlying hypergraph is a simplegraph). Therefore the computational complexity of such colorings is at least as hard as forsimple graphs.The study of conflict-free colorings was originated in the work of Even et al. [5] andSmorodinsky [13] who were motivated by the problem of frequency assignment in cellularnetworks. Specifically, cellular networks are heterogeneous networks with two different typesof nodes: base stations (that act as servers) and clients. Base stations are interconnected byan external fixed backbone network whereas clients are connected only to base stations. Con-nections between clients and base stations are implemented by radio links. Fixed frequencies∗Computer and Information Science Department, Brooklyn College, 2900 Bedford Avenue Brooklyn,NY, 11210. E-mail: amotz@sci.brooklyn.cuny.edu.†The Graduate Center, The City University of New York, 365 Fifth Avenue, New York, NY, 10016.E-mail: philaris@sci.brooklyn.cuny.edu.‡Tel-Aviv University, E-mail: olonetsk@post.tau.ac.il.§Courant Institute for Mathematical Sciences, New York University, 251 Mercer St., New York, NY, 10012.E-mail: shakhar@cims.nyu.edu.1

Page 2

are assigned to base stations to enable links to clients. Clients continuously scan frequenciesin search of a base station with good reception. The fundamental problem of frequency as-signment in such cellular networks is to assign frequencies to base stations so that every client,located within the receiving range of at least one station, can be served by some base station,in the sense that the client is located within the range of the station and no other stationwithin its reception range has the same frequency (such a station would be in “conflict” withthe given station due to mutual interference). The goal is to minimize the number of assignedfrequencies (“colors”) since the frequency spectrum is limited and costly.Suppose we are given a set of n base stations, also referred to as antennas. Assume, forsimplicity, that the area covered by a single antenna is given as a disk in the plane. Namely, thelocation of each antenna and its radius of transmission is fixed and is given (the transmissionradii of the antennas are not necessarily equal). Even et al. [5] showed that one can find anassignment of frequencies to the antennas with a total of at most O(logn) frequencies such thateach antenna is assigned one of the frequencies and the resulting assignment is free of conflicts,in the preceding sense. Furthermore, it was shown that this bound is worst-case optimal. LetR be a set of regions in the plane. For a point p ∈ ∪r∈Rr, let r(p) = {r ∈ R | p ∈ r}. Let H(R)denote the hypergraph (R,{r(p) | p ∈ ∪r∈R}). We say that H(R) is the hypergraph inducedby R. Thus, Even et al. [5] showed that any hypergraph induced by a family R of n discsin the plane admits a CF-coloring with only O(logn) colors and that this bound is tight inthe worst case. Furthermore, such a coloring can be found in deterministic polynomial time1.The results of [5] were further extended in [8] by combining more involved probabilistic andgeometric ideas. The main result of [8] is a general randomized algorithm which CF-colors anyset of n “simple” regions (not necessarily convex) whose union has “low” complexity, using a“small” number of colors. In addition to the practical motivation, this new coloring model hasdrawn much attention of researchers through its own theoretical interest and such coloringshave been the focus of several recent papers (see, e.g., [4, 5, 6, 8, 10, 12, 13, 14]). To capturea dynamic scenario where antennas can be added to the network, Fiat et al. [6] initiated thestudy of online CF-coloring of hypergraphs. They considered a very simple hypergraph Hwhich has its vertex set represented as a set P of n points on the line and its hyperedge setconsists of all intersections of the points with some interval. The set P ⊂ R is revealed by anadversary online: Initially, P is empty, and the adversary inserts points into P, one point at atime. Let P(t) denote the set P after the t-th point has been inserted. Each time a point isinserted, the algorithm needs to assign a color c(p) to it, which is a positive integer. Once thecolor has been assigned to p, it cannot be changed in the future. The coloring should remainconflict-free at all times. That is, for any interval I that contains points of P(t), there is a colorthat appears exactly once in I. Among other results, [6] provided a randomized algorithm foronline CF-coloring n points on the line with O(lognloglogn) colors with high probability.2They also provided a deterministic algorithm for online CF-coloring n points on the line withΘ(log2n) colors in the worst case.An online CF-coloring framework:of online CF-coloring applied to arbitrary hypergraphs. Suppose the vertices of an underlyinghypergraph H = (V,E) are given online by an adversary. Namely, at every given time step t, anew vertex vt∈ V is given and the algorithm must assign vta color such that the coloring is a1In [5] it is shown that finding the minimum number of colors needed to CF-color a given collection of discsis NP-hard even when all discs are congruent, and an O(logn) approximation-ratio algorithm is provided.2Their algorithm assumes that the adversary is oblivious in the sense that it does not have access to therandom coin flips of the algorithm.In this paper, we investigate the most general form2

Page 3

valid conflict-free coloring of the hypergraph that is induced by the vertices Vt= {v1,...,vt}(see the exact definition in section 2). Once vtis assigned a color, that color cannot be changedin the future. The goal is to find an algorithm that minimizes the maximum total number ofcolors used (where the maximum is taken over all permutations of the set V ).We present a general framework for online CF-coloring any hypergraph. Interestingly, thisframework is a generalization of some known coloring algorithms. For example the Unique-Max Algorithm of [6] can be described as a special case of our framework. Also, when theunderlying hypergraph is a simple graph then the First-Fit online algorithm is another specialcase of our framework.Based on this framework, we introduce a randomized algorithm and show that the maxi-mum number of colors used is a function of the ‘degeneracy’ of the hypergraph. We define thenotion of a k-degenerate hypergraph as a generalization of the same notion for simple graphs.Specifically we show that if the hypergraph is k-degenerate, then our algorithm uses O(klogn)colors with high probability. This is asymptotically tight for any constant k.As demonstrated in [6], the problem of online CF-coloring the very special hypergraphinduced by points on the real line with respect to intervals is highly non-trivial. The bestrandomized online CF-coloring algorithm of [6] uses O(lognloglogn) colors. Chen, Kaplanand Sharir [10] studied the special hypergraph induced by points in the plane with respectto halfplanes and unit discs and obtained a randomized online CF-coloring with O(log3n)colors with high probability. Recently, the bound Θ(logn) just for these two special cases wasobtained independently by Chen [3]. Our algorithm is more general and uses only Θ(logn)colors; an interesting evidence to our algorithm being fundamentally different from the ones in[3, 6, 10], when used for the special case of hypergraphs that arise in geometry, is that it usesexponentially fewer random bits. The algorithms of [3, 10] use Θ(nlogn) random coin flipsand our algorithm uses O(logn) random coin flips.Another interesting corollary of our result is that we obtain a randomized online coloring fork-inductive graphs with O(klogn) colors with high probability. This case was studied by Irani[9] who showed that a greedy First-Fit algorithm achieves the same bound deterministically.Deterministic online CF-coloring with recoloring:coloring where at each step, in addition to the assignment of a color to the newly inserted point,we allow some recoloring of other points. The bi-criteria goal is to minimize the total numberof recolorings done by the algorithm and the total number of colors used by the algorithm.We introduce an online algorithm for CF-coloring points on the line with respect to intervals,where we recolor at most one already assigned point at each step. Our algorithm uses Θ(logn)colors. This is in contrast with the O(log2n) colors used by the best known deterministicalgorithm by Fiat et al. [6] that does not recolor points. We also show online algorithm forCF-coloring points on the plane with respect to halfplanes that uses Θ(logn) colors and thetotal number of recolorings is O(n). For this problem no deterministic algorithm was knownbefore.We initiate the study of online CF-Paper organization:presents the general framework for online CF-coloring of hypergraphs. Section 4 introducesthe randomized algorithm derived from the framework. Section 5 shows deterministic onlinealgorithm for intervals and halfplanes with recoloring. Section 6 describes the results for thehypergraphs that arise from geometry. Finally, Section 7 concludes with a discussion and someopen problems.Section 2 defines the notion of a k-degenerate hypergraph. Section 33

Page 4

2PreliminariesWe start with some basic definitions:Definition 2.1. Let H = (V,E) be a hypergraph. For a subset V?⊂ V let H(V?) be thehypergraph (V?,E?) where E?= {e ∩ V?|e ∈ E}. H(V?) is called the induced hypergraph on V?.Definition 2.2. For a hypergraph H = (V,E), the Delaunay graph G(H) is the simple graphG = (V,E) where the edge set E is defined as E = {(x,y) | {x,y} ∈ E} (i.e., G is the graphon the vertex set V whose edges consist of all hyperedges in H of cardinality two).Definition 2.3. A simple graph G = (V,E) is called k-degenerate (or k-inductive) for somepositive integer k, if every (vertex-induced) subgraph of G has a vertex of degree at most k.Definition 2.4. Let k > 0 be a fixed integer and let H = (V,E) be a hypergraph on n vertices.Fix a subset V?⊂ V . For a permutation π of V?such that V?= {v1,...,vi} (where i = |V?|)let Cπ(V?) =?iAssume that ∀V?⊂ V and for all permutations π ∈ S|V?|we have Cπ(V?) ≤ k|V?|. Then wesay that H is k-degenerate.j=1d(vj), where d(vj) = |{l < j|(vj,vl) ∈ G(H({v1,...,vj}))}|, that is, d(vj) isthe number of neighbors of vjin the Delaunay graph of the hypergraph induced by {v1,...,vj}.It is easy to see that our definition of a k-degenerate hypergraph is a generalization of thatof a k-degenerate graph.3An online CF-coloring frameworkLet H = (V,E) be any hypergraph. Our goal is to define a framework that colors the verticesV in an online fashion. That is, the vertices of V are revealed by an adversary one at a time.At each time step t, the algorithm must assign a color to the newly revealed vertex vt. Thiscolor cannot be changed in the future. The coloring has to be conflict-free for all the inducedhypergraphs H(Vt) t = 1,...,n, where Vt⊂ V is the set of vertices revealed by time t.For a fixed positive integer h, let A = {a1,...,ah} be a set of h auxiliary colors (not to beconfused with the set of ‘real’ colors used for the CF-coloring: {1, 2, ...}). Let f : N → A besome fixed function. We now define the framework that depends on the choice of the functionf and the parameter h.A table (to be updated online) is maintained where each entry i at time t is associatedwith a subset Vicolors. We say that f(i) is the color that represents entry i in the table. At the beginning allentries of the table are empty. Suppose all entries of the table are updated until time t − 1and let vt be the vertex revealed by the adversary at time t. The framework first checks ifan auxiliary color can be assigned to vtsuch that the auxiliary coloring of V1the color of vtis a proper coloring of H(V1used by the framework. For example a first-fit greedy in which all colors in the order a1, ...,ahare checked until one is found. If such a color cannot be found for vt, then entry 1 is leftwith no changes and the process continues to the next entry. If however, such a color can beassigned, then vtis added to the set V1vt. If this color is the same as f(1) (the auxiliary color that is associated with entry 1), thenthe final color in the online CF-coloring of vtis 1 and the updating process for the t-th vertexstops. Otherwise, if an auxiliary color cannot be found or if the assigned auxiliary color is notthe same as the color associated with this entry, the updating process continues to the nextt⊂ Vtin addition to an auxiliary proper coloring of H(Vit) with at most ht−1together witht−1∪ {vt}). Any (proper) coloring procedure can bet−1. Let c denote such an auxiliary color assigned to4

Page 5

entry. The updating process stops at the first entry i for which vtis both added to Vithe auxiliary color assigned to vtis the same as f(i). The color of vtin the final conflict-freecoloring is then set to i.It is possible that vtnever gets a final color. In this case we say that the framework does nothalt. However, termination can be guaranteed by imposing some restrictions on the auxiliarycoloring method and the choice of the function f. For example, if first-fit is used for theauxiliary colorings at any entry and if f is the constant function f(i) = a1, for all i, then theframework is guaranteed to halt for any time t. An example instantiation of the frameworkfor conflict-free coloring with respect to intervals is given in the appendix. In section 4 wederive a randomized online algorithm based on this framework. This algorithm always halts3and moreover it halts after a “small” number of entries with high probability.We now turn to prove that the above framework produces a valid CF-coloring in case ithalts.tandLemma 3.1. If the above framework halts for any vertex vt then it produces a valid onlineCF-coloring of H.Proof. See appendix.The above algorithmic framework can also describe some well-known deterministic algo-rithms. For example, if first-fit is used for auxiliary colorings and f is the constant function,f(i) = a1, for all i, then: (a) If the input hypergraph is induced by points on a line withrespect to intervals as in example 1 then the algorithm derived from the framework becomesidentical to the Unique Maximum Greedy algorithm described and analyzed in [6]. (b) If theinput is a k-degenerate graph (also called k-inductive graph), the derived algorithm is identi-cal to the First-Fit greedy algorithm for coloring graphs online. The performance of First-Fitfor restricted classes of graphs has been analyzed in several papers [7, 9, 11]. Especially fork-inductive graphs, the First-Fit algorithm is analyzed in [9], where it is proved that it usesO(klogn) colors.4An online randomized CF-coloring algorithmThere is a randomized online CF-coloring in the oblivious adversary model that always pro-duces a valid coloring and the number of colors used is related to the degeneracy of theunderlying hypergraph in a manner described in theorem 4.1.Theorem 4.1. Let H = (V,E) be a k-degenerate hypergraph on n vertices. Then there existsa randomized online CF-coloring for H which uses at most O(log1+with high probability.14k+1n) = O(klogn) colorsThe algorithm is based on the framework of section 3. In order to define the algorithm, weneed to state what is the function f, the set of auxiliary colors of each entry and the algorithmwe use for the auxiliary coloring at each entry. We use the set A = {a1,...,a2k+1}. For eachentry i, the representing color f(i) is chosen uniformly at random from A. We use a first-fitalgorithm for the auxiliary coloring.Our assumption on the hypergraph H (being k-degenerate) implies that at least half of thevertices up to time t that ‘reached’ entry i (but not necessarily added to entry i), denoted by3It is easy to prove that in the randomized version the algorithm will halt with probability 1, but we omitthe discussion of this issue here5